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Version 6 |
| Let $f:A\to B$ be a morphism in a category $\mathcal{C}$. The \emph{kernel pair} of $f$ is defined as the pair of morphisms $(k_1: K\to A, k_2:K\to A)$ such that |
Let $f:A\to B$ be a morphism in a category $\mathcal{C}$. The \emph{kernel pair} of $f$ is defined as the pair of morphisms $(k_1: K\to A, k_2:K\to A)$ such that |
| $$\xymatrix@+=4pc{ |
$$\xymatrix@+=4pc{ |
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{K}\ar[r]^{k_1}\ar[d]_{k_2} &{A}\ar[d]^{f} \\
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{K}\ar[r]^{k_1}\ar[d]^{k_2} &{A}\ar[d]^{f} \\
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{A}\ar[r]_{f}&{B}
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{A}\ar[r]^{f}&{B}
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} |
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$$ |
| is a pullback diagram. |
is a pullback diagram. |
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| Since |
Since |
| $$\xymatrix@+=4pc{ |
$$\xymatrix@+=4pc{ |
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{A}\ar[r]^{1_A}\ar[d]_{1_A} &{A}\ar[d]^{f} \\
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{A}\ar[r]^{1_A}\ar[d]^{1_A} &{A}\ar[d]^{f} \\
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{A}\ar[r]_{f}&{B}
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{A}\ar[r]^{f}&{B}
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} |
| $$ |
$$ |
| is a commutative diagram, we have a unique morphism $g:A\to K$ such that |
is a commutative diagram, we have a unique morphism $g:A\to K$ such that |
| $$\xymatrix@+=4pc{ |
$$\xymatrix@+=4pc{ |
| A\ar@/^1ex/[rrd]^{1_A} \ar@/_1ex/[rdd]_{1_A} \ar[rd]^g & & \\ |
A\ar@/^1ex/[rrd]^{1_A} \ar@/_1ex/[rdd]_{1_A} \ar[rd]^g & & \\ |
| & K \ar[d]^{k_2} \ar[r]_{k_1} & A\ar[d]^f \\ |
& K \ar[d]^{k_2} \ar[r]_{k_1} & A\ar[d]^f \\ |
| & A\ar[r]_f & B. |
& A\ar[r]_f & B. |
| } |
} |
| $$ |
$$ |
| is commutative. As a result, $k_1$ and $k_2$ are both monomorphisms: if $k_1\circ h_1 = k_1\circ h_2$, then $$h_1 = 1_A \circ h_1 = (g\circ k_1) \circ h_1 = g\circ (k_1 \circ h_1) =g\circ (k_1 \circ h_2) = (g\circ k_1) \circ h_2 = 1_A \circ h_2 = h_2.$$ |
is commutative. As a result, $k_1$ and $k_2$ are both monomorphisms: if $k_1\circ h_1 = k_1\circ h_2$, then $$h_1 = 1_A \circ h_1 = (g\circ k_1) \circ h_1 = g\circ (k_1 \circ h_1) =g\circ (k_1 \circ h_2) = (g\circ k_1) \circ h_2 = 1_A \circ h_2 = h_2.$$ |
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| For example, in \textbf{Set}, the category of sets, the kernel pair of a function $f:A\to B$ is the pair $p_1:K\to A$ and $p_2:K\to A$, given by $$K=\lbrace (a,b) \in A\times A \mid f(a)=f(b) \rbrace,$$ and $p_1$ and $p_2$ are given by $$p_1(a,b)=a \qquad \mbox{and} \qquad p_2(a,b)=b.$$ |
For example, in \textbf{Set}, the category of sets, the kernel pair of a function $f:A\to B$ is the pair $p_1:K\to A$ and $p_2:K\to A$, given by $$K=\lbrace (a,b) \in A\times A \mid f(a)=f(b) \rbrace,$$ and $p_1$ and $p_2$ are given by $$p_1(a,b)=a \qquad \mbox{and} \qquad p_2(a,b)=b.$$ |
| This is just the kernel of a function, in the sense of universal algebra. Please see \PMlinkname{this entry}{KernelOfAHomomorphismBetweenAlgebraicSystems} for more details. |
This is just the kernel of a function, in the sense of universal algebra. Please see \PMlinkname{this entry}{KernelOfAHomomorphismBetweenAlgebraicSystems} for more details. |
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| The notion of \emph{cokernel pair} is dually defined. |
The notion of \emph{cokernel pair} is dually defined. |
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| \textbf{Remark}. $f:A\to B$ is a monomorphism iff the kernel pair of $f$ is $(1_A,1_A)$. Dually, $f$ is an epimorphism iff the cokernel pair of $f$ is $(1_A,1_A)$. |
\textbf{Remark}. $f:A\to B$ is a monomorphism iff the kernel pair of $f$ is $(1_A,1_A)$. Dually, $f$ is an epimorphism iff the cokernel pair of $f$ is $(1_A,1_A)$. |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{fb} F. Borceux \emph{Basic Category Theory, Handbook of Categorical Algebra I}, Cambridge University Press, Cambridge (1994) |
\bibitem{fb} F. Borceux \emph{Basic Category Theory, Handbook of Categorical Algebra I}, Cambridge University Press, Cambridge (1994) |
| \end{thebibliography} |
\end{thebibliography} |
